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I do not quite understand the idea an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system is integrable?

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There has been extensive discussion on this here. Infact the question was posed by Gil Kalai, a well known mathematician. I hope it helps.

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For mathematicians the question is very hard, for physicists the following jargon is used to intuitively define integrability: solvable=integrable, separable=integrable, existence of a Lax pair=integrable, # of conserved quantities equats to # of degrees of freedom = integrable.

It is interesting that the mathematicians have managed to find the exceptions to all the intuitive definitions. In general the most general physical definition is the following: if there is no of chaos then the system is integrable.

For more information look at Zakharov V.E. What is integrability (Springer, 1992).

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